Recent work [1]} has introduced the deep image prior (DIP) [2]} idea into eq:stropt: \(x\)  is reparametrized as \({\mathbf {x}}= g_{\theta }(\beta )\) , where \(g_{\theta }\)  is a deep neural network parametrized by \(\theta \) , and \(\beta \)  is a fixed random input. DIP has shown great promise for solving difficult visual inverse problems (see, e.g., discussions in [3]}, [4]}, [5]}), and can implicitly promote spatial continuity of the design—which is not explicitly modeled in eq:stroptThe state-of-the-art methods for structural optimization use smoothing filters during iteration heuristically to encourage spatial continuity; see, e.g., [6]}.. Also, the overparametrization in DIP tends to ease the global optimization. Relaxing the integer constraint \({\mathbf {x}}\in {0,1}^d\)  into a box constraint \({\mathbf {x}}\in [0,1]^d\)  as is typically done in structural optimization, we arrive at
\(\begin{split}\min _{\theta ,\mathbf {u}} \mathbf {u}^{{\mathbf {K}}(g_{\theta }(\beta ))\mathbf {u} \quad \operatorname{s.t.}{\mathbf {K}}(g_{\theta }(\beta )) \mathbf {u}=\mathbf {f},{\mathbf {V}}(g_{\theta }(\beta )) \le v_0,g_{\theta }(\beta ) \in [0,1]^d}.\end{split}In order to solve {eq:str_opt_dip}, recent work~\cite {hoyer2019neural} transforms {eq:str_opt_dip} into an unconstrained problem, which includes 1) eliminating u from the physical constraint by solving the linear system {\mathbf {K}}(g_{\theta }(\beta )) \mathbf {u}=\mathbf {f}; 2) enforcing the couple of constraints {\mathbf {V}}(g_{\theta }(\beta )) \le v_0 and g_{\theta }(\beta ) \in [0,1]^d via reparametrization, binary search, and implicit differentiation. By contrast, using \texttt {PyGRANSO}\ we can directly solve {eq:str_opt_dip} without any of the problem-specific tricks. For realistic design problems, x needs to be discrete-valued (it can have more than 2 discrete values in multi-material design), and the constraint {\mathbf {K}}(g_{\theta }(\beta )) \mathbf {u}=\mathbf {f} becomes nonlinear---K becomes a nonlinear operator acting on u due to the governing PDEs~\cite {chandrasekhar2021auto,chandrasekhar2021tounn}. \texttt {PyGRANSO}\ \,can easily handle these general cases also, whereas the tricks used in~\cite {hoyer2019neural} cannot be generalized.\) 
